2.2 Fundamentals of set theory

One of the most important notions of discrete mathematics is the notion of a set: a set is a collection of items, technically called the elements of the set. The strength of set theory is that it enables one to work with various branches of mathematical modelling under one umbrella, thus facilitating the connection between different domains, such as for example, computing and music.

Set theory is an ideal starting point for tracking problems which involve large amounts of data that need to be examined from different structural viewpoints. A typical use of set theory is for computer applications that involve large databases. Here, set theory provides efficient ways for describing the various categories of information by retrieving and combining clusters of data. For example, a single file of a database may contain the set of all pieces of music composed by a certain composer; another file may contain a collection of musical instruments, and so on. This database could be programmed to establish the relationship between the files. For instance, a relationship labelled as employed in could be established between members of the set of musical instruments and members of the set of pieces of music (Figure 2.3). Many other relationships could be specified; another example could be an orchestration rule defining which elements of the instruments set combine well in specific musical circumstances.

Figure 2.3: Set theory provides an efficient way of structuring data.

One important condition of set theory is that...